02-296 Qing-Hui LIU, Zhi-Ying WEN

Hausdorff dimension of spectrum of one-dimensional Schr\"odinger operator with Sturmian potentials (354K, Postscript) Jul 7, 02

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Abstract. Let $\beta\in(0,1)$ be an irrational, and $[a_1,a_2,\cdots]$ the continued fraction expansion of $\beta$. Let $H_\beta$ be the one-dimensional Schr\"odinger operator with Sturmian potentials. We prove that if the potential strength $V>20$, then the Hausdorff dimension of the spectrum $\sigma(H_\beta)$ is strictly great than zero for any irrational $\beta$, and is strictly less than $1$ if and only if $\liminf\limits_{k\rightarrow\infty}(a_1 a_2 \cdots a_k)^{1/k}<\infty$.

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